Quasi-constant heating at the footpoints of loops leads to evaporation and condensation cycles of the plasma: thermal non-equilibrium (TNE). This phenomenon is believed to play a role in the formation of prominences and coronal rain. However, it is often discarded to be involved in the heating of warm loops as the models do not reproduce observations. Recent simulations have shown that these inconsistencies with observations may be due to oversimplifications of the geometries of the models. In addition, our recent observations reveal that long-period intensity pulsations (several hours) are common in solar coronal loops. These periods are consistent with those expected from TNE. The aim of this paper is to derive characteristic physical properties of the plasma for some of these events to test the potential role of TNE in loop heating. We analyzed three events in detail using the six EUV coronal channels of SDO/AIA. We performed both a Differential Emission Measure (DEM) and a time-lag analysis, including a new method to isolate the relevant signal from the foreground and background emission. For the three events, the DEM undergoes long-period pulsations, which is a signature of periodic heating even though the loops are captured in their cooling phase, as is the bulk of the active regions. We link long-period intensity pulsations to new signatures of loop heating with strong evidence for evaporation and condensation cycles. We thus witness simultaneously widespread cooling and TNE. Finally, we discuss the implications of our new observations for both static and impulsive heating models. ; Comment: 40 pages, 30 figures, accepted for publication in the Astrophysical Journal, v2: corrected typos